Switching-based Sinusoidal Disturbance Rejection for Uncertain StableLinear Systems

Yang Wang, Gilberto Pin, Andrea Serrani and Thomas Parisini

Abstract— The problem of rejection of sinusoidal distur-bances with known frequencies acting on an unknown single-input single-output linear system is addressed in this note.We present a new approach that does not require knowledgeof the frequency response of the transfer function over thefrequency of interest. The proposed methodology reposes uponthe combination of the classic feedforward control algorithmand logic-based switching. The use of three different switchinglogics is proposed in this paper, namely: pre-routed, dwell-timeand hysteresis switching. A comparative evaluation of the threeswitching strategies is performed via a simulation study.

I. INTRODUCTION AND PROBLEM FORMULATION

Rejection of harmonic disturbances occurring in a controlsystem remains a central theme in control, fueled by a largenumber of technological applications, from vibration sup-pression [1] to active noise control [2], to wave attenuation inmarine systems [3]. A common thread across the different ap-proaches pursued by the control community to the solution ofvarious manifestations of the harmonic disturbance rejectionproblem is the ubiquitous internal model principle [4], whichprescribes that a suitable copy of the system generating thedisturbance must be embedded in the controller to ensurerobust regulation in the presence of model uncertainties.While the harmonic disturbance rejection problem is sub-sumed by the more general output regulation problem [5], inits prototypical form for SISO LTI plant models, the formerproblem is cast into the following setup

x = A(µ)x+B(µ)[ d(t)− d(t)] , x(0) = x0 ∈ Rr

y = C(µ)x (1)

where (1) is an r-dimensional realization of the internallystable interconnection of an uncertain plant model and arobust stabilizer. System (1) is driven by the differencebetween a sinusoid of known frequency, ω? ∈ R>0

d(t) = ψ1 cos(ω?t) + ψ2 sin(ω?t) (2)

and an estimate of the disturbance, d(t) ∈ R, generated bya feedforward regulator. The vectors µ ∈ Rp and ψ :=col(ψ1, ψ2) ∈ R2 collect the uncertain parameters of theplant model and the disturbance, respectively. It is assumedthat µ ranges on a given known compact set, P ⊂ Rp. Forfuture use, we let Wµ(s) := C(µ)(sI−A(µ))−1B(µ) denote

Yang Wang and Thomas Parisini are with the Department of Electricaland Electronic Engineering, Imperial College, London UK. Gilberto Pinis with Electrolux Professional, S.p.A., Italy; Andrea Serrani is withthe Department of Electrical and Computer Engineering, The Ohio StateUniversity, Columbus OH – USA. Corresponding author: A. Serrani,serrani.1@osu.edu

the transfer function of system (1). System (1) is assumedto be internally stable, robustly with respect to µ ∈ P:

Assumption 1.1: There exist constants a1, a2 > 0 suchthat the parameterized family Px : Rp → Rr×r of solutionsof the Lyapunov equation Px(µ)A(µ) +AT (µ)Px(µ) = −Isatisfies a1I ≤ Px(µ) ≤ a2I for all µ ∈ P . /

The sinusoidal disturbance in (2) can be represented as theoutput of an exosystem of the form

ω = Sω, ω(0) = ω0 = ψ ∈ R2

d = Γω (3)

where S =

(0 ω?

−ω? 0

), Γ = (1 0). As a consequence,

the original problem is cast into the framework of outputregulation [4], [6], with the additional assumption of internalstability of the plant model:

Problem 1: (Output Regulation Problem) For system (1),design a dynamic output-feedback controller of the form

ξ = fa(ξ, y) , ξ(0) = ξ0 ∈ Rm

d = ha(ξ, y), (4)

with fa(·, ·) : Rm × R 7→ Rm and ha(·, ·) : Rm × R 7→ R,such that the trajectories of the closed-loop system (1), (3)and (4) originating from all initial conditions x0 ∈ Rr, ω0 ∈R2 and ξ0 ∈ X , where X ⊂ Rm is a set to be determined,are bounded and satisfy limt→∞ y(t) = 0 for all µ ∈ P . /

Necessary and sufficient conditions of solution of Problem 1have been known for a long time [5]–[7]. Specialized to thecase of SISO systems, these conditions are hereby assumed:

Assumption 1.2: (i) System (1) is controllable and observ-able for any µ ∈ P; (ii) Over the range of frequencies ofinterest, Wµ(jω?) 6= 0 for all µ ∈ P .

Early work in the realm of adaptive feedforward con-trol (AFC) assume knowledge of Re{Wµ(jω?)} andIm{Wµ(jω?)} as a prerequisite for the controller design [1],[8]–[10], such a condition is usually termed an SPR-likecondition. Subsequent works have attempted an adaptiveestimation of said quantities within AFC schemes [11],[12]; however, issues related to asymptotic convergence andinteraction with the plant dynamics were left open. Therecent contribution [13], has shown that given the knowledgeof either sign Re{Wµ(jω?)} or sign Im{Wµ(jω?)}, one orboth of the two following dynamic feedback controllers withd = ω1 solves the Problem 1 for sufficient small values of

Plant

Stabilizing Controller

Supervisor

𝑢𝑐𝑦𝑐

Stabilized PlantExosystem

−𝑑

𝑢

Controller 1

Controller 2

Controller n

…

𝜂

መ𝑑

መ𝑑𝑛

መ𝑑2

መ𝑑1

𝑦

Fig. 1. Multi-Controller set up for output regulation problem.

the controller gain k > 0:{˙ω1 = ω?ω2 − ksign Re{Wµ(jω?)}y˙ω2 = −ω?ω1

(5)

{˙ω1 = ω?ω2

˙ω2 = −ω?ω1 − ksign Im{Wµ(jω?)}y(6)

In [14], the authors have proposed an adaptive controlscheme that completely removes the necessity of knowing apriori the sign of the real and imaginary parts of Wµ(jω?).The controller in [14] is characterized by a relatively highdimensionality, due to the necessity to employ a multiplemodel adaptive control scheme. In this paper, we propose analternative design, where multiple-model adaptive techniquesare replaced by switching mechanisms among fixed con-trollers. Several different switching strategies are proposed,within a common baseline control architecture. It is shownthat the proposed approach removes some of the pitfalls ofthe multiple-model adaptive controller of [14], albeit stillrequiring a relatively large-dimensional state space for theoverall controller.

The paper is organized as follows: The structure of themulti-controller employed in this paper is given in Section II.In Section III, a brief overview of the switching strategiesrelevant to the approach pursued in this paper is presented,based on the lucid exposition found in [15]. The analysisof the proposed design is presented in Section IV, whereasa comparative simulation study is reported in Section V.Concluding remarks are offered in Section VI.

Notation: We denote with ‖ · ‖ both the Euclidean vectornorm and the corresponding induced matrix norm.

II. STATE-SHARING MULTI-CONTROLLER

The overall control architecture, depicted in Fig.1, fol-lows the general paradigm of state-shared multi-controllersproposed by Morse [15] [16]. The control signal appliedto the plant is d(t) := dη(t)(t), where η : [0,∞) 7→ I isa piecewise-constant switching signal taking values in theindex set of the family of the candidate controllers I :={1, 2, · · · , n}. The system that generates the switching signalis referred to as the supervisory system. In the architectureof Fig. 1, the candidate controllers Ci, i ∈ I, are selected as

˙wi = Swi − kϑiy , wi(0) = wi0

di = Γwi , i ∈ I (7)

where ϑi is a constant estimate of the plant responseat the frequency of excitation, ϑ(µ) :=

(ϑ1 ϑ2

)T=(

Re{Wµ(jω?)} Im{Wµ(jω?)})T

. The parameter ϑ ∈ R2 isunknown, but assumed to range in a known set. Specifically,let the set Θ ⊂ R2 be the annular region defined, for givenreal numbers 0 < δ1 < δ2, as

Θ :={ϑ ∈ R2 | δ21 ≤ ϑ21 + ϑ22 ≤ δ22

}(8)

and consider the following assumption:Assumption 2.1: The unknown parameter vector ϑ(µ) sat-

isfies ϑ(µ) ∈ intΘ for all µ ∈ P . /

It is noted that the set Θ is not convex. The family of thecandidate controllers is designed in such a way that ϑi ∈ Θfor all i ∈ I, and there exists at least one controller Cj , j ∈ Iwith parameter estimate ϑj satisfying

||ϑj − ϑ|| ≤ ρ, (9)

where ρ is a constant to be selected. In the following sections,it will be shown that, for an appropriate choice of ρ andk, (9) ensures that there exists a subset of the family ofthe candidate controllers, with index set denoted by I? thatsolves the output regulation problem for any µ ∈ P .

To reduce the dimension of the overall controller, a dif-ferent parameterization of the controller is first employed byway of the coordinate change wic := T ic w

i, i ∈ I, where

T ic :=1

||ϑi||

(ϑi1 −ϑi2ϑi2 ϑi1

)and ϑij , j = 1, 2 represents the j-th element of the vectorϑi. This yields the set of state-sharing controllers [15](equivalent to (7))

˙wc = Swc − kGcy , wc(0) = wc0 ∈ R2

di = θiT wc , i ∈ I (10)

with Gc = ΓT and θi =(ϑi1 −ϑi2

)T. Since each member

of (10) has the same structure of either (5) or (6), theexistence of a controller that solves the problem is guaranteedfor a suitable value of k. Also, note that (10) reducesthe dimension of the controller from 3n to 2 + n for asingle frequency. Clearly, Assumption 2.1 holds for the re-parameterized vector θ :=

(ϑi1 − ϑi2

)T, whereas condition

(9) needs to be revised for the re-parameterized estimates θi.Remark 2.1: While the multi-controller (10) is used for

the implementation of the algorithm, we sometimes refer toits equivalent form for the ease of analysis (7).The control problem is thus reformulated as follows:

Problem 2: (Switching Logic Design Problem) For sys-tem (1), (3) and (10), design a supervisor that generatesa suitable switching signal η with a selection for k > 0and ρ > 0 such that trajectories of the closed-loop systemoriginating from any x0 ∈ Rr, ω0 ∈ R2 and ωc0 ∈ R2 arebounded and satisfy limt→∞ y(t) = 0 for all µ ∈ P . /

Σ𝜎 𝛽𝑦 𝜎 𝜂

Supervisor system

Fig. 2. Non-estimator based supervisor.

Start: 𝜎 = 1

Timer: 𝜏 = 0ሶ𝜏 = 1

Performance: J = 0 , ሶ𝐽 = 𝑦2

𝜏 = 𝜏𝐷?No

ሶ𝜏 = 0𝐽𝑃 = 𝐽

Yes

𝐽 > 𝜎𝐽𝑃?No

𝜎 = 𝜎 + 1

Yes

Fig. 3. Flowchart of the scheduling function Σσ in Fig. 2

III. SWITCHING LOGIC

This section presents the development of the supervisorsystem in Fig. 1 by using three different switching logics:Pre-routed, Hysteresis and Dwell-time switching [15], [17].

A. Non-estimator-based Supervisor: Pre-routed Switching

The non-estimator-based supervisor is the cascade con-nection of a scheduling logic Σσ and a routing functionβ(·) : {1, 2, 3, · · · ,+∞} 7→ I, as shown in Fig. 2. Theoutput σ(·) : [0,+∞) 7→ Z+ is a piecewise-constant signalto be determined. The routing function β(·) is constructedto fulfill the following revisitation property [15]:

Property 3.1 (Revisitation Property): For any q ∈ I andany i ∈ N, there exists an integer j ≥ i at which β(j) = q.Essentially, along the pre-routed path β(σ) each candi-date controller is revisited infinitely often. Here, we defineβ(σ) := mod(σ, n) + 1. The flow chart of the schedulinglogic Σσ is given in Fig. 3, where τD is a predefined timeconstant. During the first τD time units after a switch isinitiated, τ is increased form 0 to τD, while J(t) evolvesaccording to

J(t) = (y(t))2. (11)

At the end of this interval of time, the timer τ is turned offand JP is set to be equal to the present value of J . So longas J(t) remains smaller or equal than σJP , J(t) is updatedcontinuously according to (11). If and when J(t) > σJP , σis incremented by 1, meanwhile τ, J and JP are set to zero,and the entire process is repeated. Note that, thanks to τD,infinitely fast switching is avoided and the solutions to thedifferential equations involved exist and are unique.

B. Estimator-based Supervisor

Alternative switching strategies considered herein (hys-teresis and dwell-time) require a multi-estimator controllerarchitecture as shown in Fig. 4. In this scheme, ΣS representsthe switching logic, whose function is to determine η based

Estimator 1 Σ𝐽1

Estimator 2

Estimator n

Σ𝐽2

Σ𝐽𝑛

Σ𝑆… … 𝜂(𝑦, መ𝑑)

𝑦1

𝑦𝑛

𝑦2 𝐽2

𝐽1

𝐽𝑛

Fig. 4. Estimator-based supervisor system.

Start: 𝜂 =argmin𝑖{𝐽𝑖}

∃ 𝑗:1 + ℎ 𝐽𝑗 ≤ 𝐽𝜂?

𝜂 = 𝑗

YesNo

Fig. 5. Flowchart of the hysteresis switching logic.

on Ji, i ∈ I. Following [14], let Π(µ) ∈ Rr×2 be the uniquesolution of the Sylvester equation Π(µ)S = A(µ)Π(µ) +B(µ)Γ and change coordinates in (1) and (7) as ζ := wη−wand z := x−Π(µ)ζ to obtain

z = A(µ)z + kΠ(µ)ϑηy , z(0) = z0 ∈ Rr

ζ = Sζ − kϑηy , ζ(0) = ζ0 ∈ R2

y = C(µ)z + ϑT (µ)ζ , (12)

where ϑT(µ)=C(µ)Π(µ). To define the performance indexfor each candidate controller, we design the adaptive observer

˙ζ = Sζ − kϑηy − εϑη(ϑηT ζ − y) , ζ(0) = ζ0 ∈ R2

yi = ϑiT ζ , i ∈ I (13)

where ε ∈ R>0. For each i ∈ I, the performance signalgenerator Σi

J is defined as Ji = −λJi+(yi)2 , with Ji(0) =Ji0, y := yi − y and λ ∈ R>0 is a forgetting factor. GivenJi, the design of the supervisor system is completed by theselection of a suitable switching logic ΣS . In what follows,we give a brief account of the two switching strategies.

1) Hysteresis Switching [17]: The mechanism behindhysteresis switching is shown in Fig. 5. Let h ∈ R>0,called the hysteresis constant, be given. Assume that thatat certain time tn, the value of η(tn) switches to somej ∈ I. Then the value of η is kept constant until the timetn+1 ≥ tn when (1 + h) min

i∈I{(Ji(tn+1)} < Jj(tn+1), at

which point η(tn+1) is set as η(tn+1) = arg mini∈I{Ji(tn+1)}.

Repeating the above steps, a piecewise-constant signal η isgenerated, which is continuous from the right. The selectionJi0 > 0 avoids infinitely fast switching in the initializationof the algorithm. Possible non-uniqueness in the selectionof arg min

i∈I{Ji(tn+1)} is resolved by an arbitrary assignment

among the available choices.

Start: 𝜂 =argmin

𝑖{𝐽𝑖}

Timer: 𝜏 = 0ሶ𝜏 = 1

𝜏 = 𝜏𝐷?No

Yes

∃ 𝑗:𝐽𝑗 < 𝐽𝜂? No

𝜂 = j

Yes

Fig. 6. Flowchart of the Dwell-time switching logic.

2) Dwell Time Switching [18]: the mechanism is il-lustrated in Fig.6. Let τD > 0 be a chosen dwell-timeconstant, and assume that at a time tn, η switches tosome j ∈ I. The value of η is then kept constant untila time tn+1 ≥ tn occurs such that tn+1 − tn ≥ τDand min

i∈I{(Ji(tn+1)} < Jj(tn+1). At that point, η(tn+1)

is set as η(tn+1) = arg mini∈I{Ji(tn+1)}. Similarly to the

hysteresis switching, when arg min is not unique, η can bearbitrarily selected among those available choices. τD avoidsthe occurrence of infinitely fast switching.

IV. STABILITY ANALYSIS

In this section, we provide a concise treatment of the sta-bility and convergence analysis for the multi-controller (10)(equivalently, (7)) for each of the considered switching logic.To begin, we establish properties of a certain Lyapunov func-tion candidate that are instrumental in the ensuing analysis.Details are given in [14].

Property 4.1: There exist a scalar k > 0 and constants0 < c1 < c2 ≤ c3 such that the solution Po : (ϑ, k) 7→ R2×2

of the parameterized family of Lyapunov equations

Po[S−k ϑϑT

]+[S−k ϑϑT

]TPok) = −k ϑTϑI (14)

satisfies c1I ≤ Po(ϑ, k) ≤ c2I and ‖Po(ϑ, k)‖ ≤ c3 for all(ϑ, k) ∈ Θ × [0, k].

Let the time sequence {Tn}Nn=1 denote the instants at whichswitching takes place, and consider in any time interval t ∈[Tn−1, Tn) the LTV system (12), which can be seen as thefeedback interconnection of the two linear systems :

Σ1 : ζ =[S − k ϑϑT

]ζ − kϑηϑT ζ − kϑην1, y1 = θT ζ

Σ2 : z = Az + kΠθηCz + kΠθη ν2, y2 = Cz

with the interconnection structure ν1 = y2, ν2 = y1 andϑη := ϑη − ϑ. For each system Σi, i = 1, 2, the followingintermediate results hold, due to the fact that there exists atleast one candidate controller with θi verifying (9):

Lemma 4.2: There exist scalars γ?1 > 0, ρ? > 0 and k?1 ∈(0, k] such that system Σ1 is strictly dissipative with respectto the supply rate q1(ν1, η1) = γ?1

2|ν1|2 − |y1|2 for all k ∈(0, k?1) and ||ϑη|| ≤ ρ?, with quadratic, positive definite anddecrescent storage function V1(ζ) = 2k−1ζTPo(θ, k)ζ. /

Lemma 4.3: There exist scalars γ?2 > 0 and k?2 ∈ R>0

such that Σ2 is strictly dissipative with respect to the supplyrate q2(ν2, η2) = k2γ?2

2|ν2|2 − |y2|2 for all k ∈ (0, k?2),with quadratic and positive definite storage function V2(z) =zTPx(µ)z. /

The proofs of the lemmas follow from elementary Lyapunovarguments. It is worth noticing that, since the interconnectedsystem is globally Lipschitz uniformly in t, solutions existuniquely on [0,∞).

Non-estimator based controller

Next, we establish the stability property of the overallsystem with the non-estimator based switching mechanism.

Theorem 4.4: Given system (1), there exist scalars k? ∈R>0 and ρ? ∈ R>0 such that, for all the state-sharing mulitcontroller (10) with k ∈ (0, k?) and ρ ∈ (0, ρ?), the non-estimator based supervisor system described in Section III-Asolves the problem defined in Problem 2.

Proof: Applying Lemmas 4.2 – 4.3 and combiningthe L2 gains of the single subsystems, Letting k? :=min{k?1 , k?2 , (γ?1γ?2)−1}, it follows that system (12) is asmall-gain interconnection (with respect to the L2-norm) forall k ∈ (0, k?). Since ρ ∈ (0, ρ?), there exists a subset of thefamily of the candidate controllers, with index set denotedby I?, whose members solve the output regulation problem.This implies that y ∈ L2 if η ∈ I?. Assume that at t = Tmη(t) switches to some q ∈ I?. As shown in [15, Section4.2], there exists an integer σ(q) > 0, which depends onlyon the parameters of the controller, satisfying

J(t) =

∫ t

Tm

(y(τ))2dτ ≤ σ(q)

∫ Tm+τD

Tm

(y(τ))2dτ (15)

for all t ∈ [Tm, Tm+1). Resorting to the revisitation Property3.1, and due to the fact that σ will increment by 1 aftereach switch, there always exists an instant Tm at whichσ(Tm) > σ(q). This indicates that no more switching willoccur beyond this point and Tm+1 = +∞. Consequently,limt→∞ y(t) = 0.

Estimator-based controllers

For the estimator-based supervisor system, the dynamicsof the observation error, ζ := ζ − ζ, are given by

˙ζ = (S − εϑηϑηT )ζ − εϑη(ϑηT ζ − C(µ)z) (16)

Resorting to Property 4.1, one can verify that there ex-ists a positive definite symmetric matrix-valued functionPe : (ϑη, ε) 7→ R2×2 solving the parametrized family of

Lyapunov equations Pe

[S− εϑηϑηT

]+[S−εϑηϑηT

]TPe =

−ε ϑηT ϑηI. Over any time interval t ∈ [Tn−1, Tn), theoverall system can be written in the following form

ξ = F (k, ε, ϑη)ξ +Hθuη, y = Cpξ (17)

with state ξ :=(ζT zT ζT

)Tand input error uη := ϑηT ζ.

In (17), H is a constant matrix of suitable dimension, andCp = [ϑT , C(µ), 0, 0] ∈ Rr+4. Using arguments similar

0 10 20 30 40 50 60 70 80 90 100-4

-2

0

2

4

0 5 10 15 20 25 30 35 40 45 500

2

4

Fig. 7. System response using the pre-routed switching mechanism

0 10 20 30 40 50 60 70 80 90 100-4

-2

0

2

4

0 5 10 15 20 25 30 35 40 45 500

2

4

Fig. 8. System response using the hysteresis switching mechanism

to those employed in the proof of Theorem 4.4, it canbe verified that F (k, ε, ϑη) is a Hurwitz matrix for anyk ∈ (0, k?) and ||ϑη|| ≤ ρ?.

The estimator-based switching scheme consists in mon-itoring the performance indexes at each instant. After aswitch has occurred, a period called waiting time of lengthτmin is allowed to elapse before the next switch can takeplace. The waiting time is essential to prevent arbitrarilyfast switching. Note that the waiting time for the dwell-timeswitching is equal to τD, whereas for the hysteresis switchingthe waiting time is proportional to the hysteresis constant h.The following theorem establishes the stability properties ofthe closed-loop system with estimator-based switching logic:

Theorem 4.5: For the system (17) and the switching logicdescribed in Section III-B, there exist positive numbersτs, k?, ρ? and a function ρs(µ, τmin) > 0 such that ifthe waiting time τmin satisfies τmin ∈ (0, τs) and ρ ≤min{ρ?, ρs(µ, τmin)}, then the trajectories of the closed-loop system originating from any initial conditions x0 ∈ Rr,ω0 ∈ R2, ω0 ∈ R2 and ζ0 ∈ R2 are bounded and satisfylimt→∞ y(t) = 0 for all µ ∈ P and k ∈ (0, k?).The proof is carried out by contradiction and can be foundin [19]. Given τs, one can accordingly choose the waitingtime τmin, then select τD = τmin for the dwell-time logicand a sufficiently small h for the hysteresis switching suchthat the condition τmin ∈ (0, τs) is verified.

V. ILLUSTRATIVE EXAMPLE

In this section, we provide a simulation study to showthe effectiveness of the proposed algorithms and com-

0 10 20 30 40 50 60 70 80 90 100-4

-2

0

2

0 5 10 15 20 25 30 35 40 45 500

2

4

Fig. 9. System response using the dwell-time switching mechanism

0 50 100 150 200-5

0

5

0 50 100 150 2000

2

4

Fig. 10. System response with additional input noise: Pre-routed switching

pare the performance of the three different switching log-ics. Consider the stable non-minimum phase plant modelW (s) = 2(s−1)

s2+2s+5 , and let the disturbance signal be givenby d(t) = 2 sin(2t). The frequency response of the plantat the frequency of excitation yields the parameter vectorθ = (0.8235,−0.7059)T . The family of the candidatecontrollers are chosen as I := {1, 2, 3, 4} with correspondingparameter estimates : θ1 = (1, 0)T , θ2 = (0,−1)T , θ3 =(−1, 0)T , θ4 = (0, 1)T . Note that the considered familycorresponds to four different controllers of the type proposedin [13]. To examine the worst case scenario, we choosethe third controller as the initial controller, whose parameterestimate is located at the furthest location from the true value.The gain parameters for all three switching mechanisms arek = 0.5, τD = 1, h = 0.35, ε = 0.1, λ = 1e− 3.

The time history of the plant output and that of theswitching signal are reported in Fig. 7 through Fig. 9for each of the three switching logic. It is observed thatthe proposed multiple controller with all three switchingmechanisms succeed in cancelling the periodic disturbancewith comparable convergence speed and transient behavior.It is seen that each switching logic is capable of selecting themost appropriate controller (θ1 in this case). It is noted thatthe hysteresis switching approach can be tuned to achievea faster response; however, the tuning parameters have beenpurposefully selected to enforce a similar convergence speedfor all three systems to facilitate a comparative analysis.

Next, the robustness of the switching mechanisms is testedby adding additive noise at the plant input. Specifically,the sinusoidal disturbance is replaced by the signal d(t) =

0 50 100 150 200-5

0

5

0 50 100 150 2000

5

Fig. 11. System response with additional input noise: Dwell-time switching

0 50 100 150 200-5

0

5

0 50 100 150 2000

2

4

Fig. 12. System response with additional input noise: Hysteresis switching

2 sin(2t) + n(t), where n(t) is uniformly distributed in[−2, 2]. Note that this is a stringent test, as the magnitude ofthe noise is comparable to that of the sinusoidal signal. Thesame selection of controller parameters has been adoptedfor the simulations. As seen in Fig. 10 and Fig. 11, thebehavior the pre-routed and dwell-time switching controllerhas worsened considerably due to the effect of the inputnoise, especially for the pre-routed scheme. On the otherhand, the hysteresis-based switching method demonstratesbetter robustness properties.

VI. CONCLUDING REMARKS

A state-sharing multi-controller architecture has been pro-posed to overcome the necessity of knowing the parametersrelated to the frequency response of the stabilized plant(or their sign) in the adaptive feedforward approach tothe harmonic disturbance rejection problem. The rationalebehind the method is to design a sufficient large number ofmultiple model-based controllers such that there exist modelswith constant parameter estimates that are sufficiently closeto the unknown parameter vector of the plant. Then, using asuitable performance criterion and proper switching logic,one can eventually find the most appropriate model, andconsequently activate the corresponding controller.

The obvious drawback of the algorithm is its low effi-ciency due to the possible high dimension of the multiplemodels, notwithstanding the reduction allowed by using astate-shared technique. While in the single harmonic casethe order of these controllers compares very favorably withthe order of the multiple-model adaptive controller proposedin [14], this advantage disappears quickly in a more general

case when multiple harmonics are considered. At this pointit is unclear whether the switching strategies proposed hereare advantageous from the dimensionality standpoint versusa multi-frequency version of the adaptive controller proposedin [14]. It appears that a judicious combination of the methodproposed in this paper with the multiple-model adaptivecontrol of [14] is perhaps the winning strategy to obtain a“universal” adaptive feedforward regulator with the smallestpossible dimension. This avenue is currently being pursedand extended to the case of multiple harmonics.

REFERENCES

[1] K. Ariyur and M. Krstic, “Feedback attenuation and adaptive can-cellation of blade vortex interaction on a helicopter blade element,”IEEE Transactions on Control Systems Technology, vol. 7, no. 5, pp.596–605, 2002.

[2] S. M. Kuo and D. R. Morgan, “Active noise control: A tutorial review,”Proceedings of the IEEE, vol. 87, no. 6, pp. 943–973, 1999.

[3] H. I. Basturk and M. Krstic, “Adaptive wave cancelation by acceler-ation feedback for ramp-connected air cushion-actuated surface effectships,” Automatica, vol. 49, no. 9, pp. 2591 – 2602, 2013.

[4] B. Francis and W. Wonham, “The internal model principle of controltheory,” Automatica, vol. 12, no. 5, pp. 457 – 465, 1976.

[5] B. A. Francis, “The linear multivariable regulator problem,” SIAMJournal on Control and Optimization, vol. 15, no. 3, pp. 486–505,1977.

[6] E. Davison, “The robust control of a servomechanism problem forlinear time-invariant multivariable systems,” IEEE Transactions onAutomatic Control, vol. 21, no. 1, pp. 25–34, Feb 1976.

[7] B. A. Francis and W. M. Wonham, “The role of transmission zerosin linear multivariable regulators,” International Journal of Control,vol. 22, no. 5, pp. 657–681, nov 1975.

[8] M. Bodson, A. Sacks, and P. Khosla, “Harmonic generation in adaptivefeedforward cancellation schemes,” IEEE Transactions on AutomaticControl, vol. 39, no. 9, pp. 1939–1944, 1994.

[9] W. Messner and M. Bodson, “Design of adaptive feedforward al-gorithms using internal model equivalence,” International Journal ofAdaptive Control and Signal Processing, vol. 9, no. 2, pp. 199–212,1995.

[10] M. Bodson, J. Jensen, and S. Douglas, “Active noise control for peri-odic disturbances,” IEEE Transactions on Control Systems Technology,vol. 9, no. 1, pp. 200–205, oct 2001.

[11] J. Chandrasekar, L. Liu, D. Patt, P. Friedmann, and D. Bernstein,“Adaptive Harmonic Steady-State Control for Disturbance Rejection,”IEEE Transactions on Control Systems Technology, vol. 14, no. 6, pp.993–1007, nov 2006.

[12] S. Pigg and M. Bodson, “Adaptive Algorithms for the Rejection of Si-nusoidal Disturbances Acting on Unknown Plants,” IEEE Transactionson Control Systems Technology, vol. 18, no. 4, pp. 822–836, 2010.

[13] R. Marino and P. Tomei, “Output Regulation for Unknown StableSystems,” IEEE Transactions on Automatic Control, vol. 60, no. 8,pp. 2213–2218, 2015.

[14] Y. Wang, G. Pin, A. Serrani, and T. Parisini, “Removing SPR-likeconditions in adaptive feedforward control of uncertain systems,” inProceedings of the IEEE Conference on Decision and Control, LasVegas, NV, 2016, pp. 4728–4733.

[15] A. S. Morse, “Control using logic-based switching,” in Trends inControl. A European Perspective. Springer Verlag, 1995, pp. 69–113.

[16] S. Fujii, J. P. Hespanha, and A. S. Morse, “Supervisory control offamilies of noise suppressing controllers,” in Proceedings of the 37thIEEE Conference on Decision and Control, vol. 2, 1998, pp. 1641–1646.

[17] J. P. Hespanha, D. Liberzon, and A. Morse, “Hysteresis-based switch-ing algorithms for supervisory control of uncertain systems,” Automat-ica, vol. 39, no. 2, pp. 263 – 272, 2003.

[18] M. Cao and A. S. Morse, “Dwell-time switching,” Systems and ControlLetters, vol. 59, no. 1, pp. 57–65, 2010.

[19] K. Narendra and J. Balakrishnan, “Adaptive control using multiplemodels,” IEEE Transactions on Automatic Control, vol. 42, no. 2, pp.171–187, 1997.